The Clowder Project

The category $\mathsf{Sets}$ admits a closed symmetric bimonoidal category structure consisting of:

• The Underlying Category. The category $\mathsf{Sets}$ of pointed sets.
• The Additive Monoidal Product. The coproduct functor
  \[ \mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\colon \mathsf{Sets}\times \mathsf{Sets}\to \mathsf{Sets} \]

  of Chapter 2: Constructions With Sets, Item 1 of Proposition 2.2.3.1.3.

• The Multiplicative Monoidal Product. The product functor
  \[ \times \colon \mathsf{Sets}\times \mathsf{Sets}\to \mathsf{Sets} \]

  of Chapter 2: Constructions With Sets, Item 1 of Proposition 2.1.3.1.3.

• The Monoidal Unit. The functor
  \[ \mathbb {1}^{\mathsf{Sets}} \colon \mathsf{pt}\to \mathsf{Sets} \]

  of Definition 3.1.3.1.1.

• The Monoidal Zero. The functor
  \[ \mathbb {0}^{\mathsf{Sets}} \colon \mathsf{pt}\to \mathsf{Sets} \]

  of Definition 3.1.3.1.1.

• The Internal Hom. The internal Hom functor
  \[ \mathsf{Sets}\colon \mathsf{Sets}^{\mathsf{op}}\times \mathsf{Sets}\to \mathsf{Sets} \]

  of Chapter 2: Constructions With Sets, of .

• The Additive Associators. The natural isomorphism
  \[ \alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}} \colon {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\circ {\webleft ({\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\times \text{id}_{\mathsf{Sets}}\webright )} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\circ {\webleft (\text{id}_{\mathsf{Sets}}\times {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\webright )}\circ {\mathbf{\alpha }^{\mathsf{Cats}}_{\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}} \]

  of Definition 3.2.3.1.1.

• The Additive Left Unitors. The natural isomorphism
  \[ \lambda ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\colon {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\circ {\webleft (\mathbb {0}^{\mathsf{Sets}}\times \text{id}_{\mathsf{Sets}}\webright )} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathbf{\lambda }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}} \]

  of Definition 3.2.4.1.1.

• The Additive Right Unitors. The natural isomorphism
  \[ \rho ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\colon {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\circ {\webleft ({\mathsf{id}}\times {\mathbb {0}^{\mathsf{Sets}}}\webright )}\mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathbf{\rho }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}} \]

  of Definition 3.2.5.1.1.

• The Additive Symmetry. The natural isomorphism
  \[ \sigma ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}} \colon {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\circ {\mathbf{\sigma }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets},\mathsf{Sets}}} \]

  of Definition 3.2.6.1.1.

• The Multiplicative Associators. The natural isomorphism
  \[ \alpha ^{\mathsf{Sets}} \colon {\times }\circ {\webleft ({\times }\times \text{id}_{\mathsf{Sets}}\webright )} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}{\times }\circ {\webleft (\text{id}_{\mathsf{Sets}}\times {\times }\webright )}\circ {\mathbf{\alpha }^{\mathsf{Cats}}_{\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}} \]

  of Definition 3.1.4.1.1.

• The Multiplicative Left Unitors. The natural isomorphism
  \[ \lambda ^{\mathsf{Sets}}\colon {\times }\circ {\webleft (\mathbb {1}^{\mathsf{Sets}}\times \text{id}_{\mathsf{Sets}}\webright )} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathbf{\lambda }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}} \]

  of Definition 3.1.5.1.1.

• The Multiplicative Right Unitors. The natural isomorphism
  \[ \rho ^{\mathsf{Sets}}\colon {\times }\circ {\webleft ({\mathsf{id}}\times {\mathbb {1}^{\mathsf{Sets}}}\webright )}\mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathbf{\rho }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}} \]

  of Definition 3.1.6.1.1.

• The Multiplicative Symmetry. The natural isomorphism
  \[ \sigma ^{\mathsf{Sets}} \colon {\times } \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}{\times }\circ {\mathbf{\sigma }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets},\mathsf{Sets}}} \]

  of Definition 3.1.7.1.1.

• The Left Distributor. The natural isomorphism

  \[ \delta ^{\mathsf{Sets}}_{\ell } \colon \mathord {\times }\circ \webleft (\text{id}_{\mathsf{Sets}}\times \mathord {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\webright ) \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathord {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\circ \webleft (\mathord {\times }\times \mathord {\times }\webright )\circ \mathbf{\mu }^{\mathsf{Cats}_{\mathsf{2}}}_{4|\mathsf{Sets},\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}\circ \webleft (\Delta _{\mathsf{Sets}}\times \webleft (\text{id}_{\mathsf{Sets}}\times \text{id}_{\mathsf{Sets}}\webright )\webright ) \]

  of Definition 3.3.1.1.1.
• The Right Distributor. The natural isomorphism

  \[ \delta ^{\mathsf{Sets}}_{r} \colon \mathord {\times }\circ \webleft (\mathord {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\times \text{id}_{\mathsf{Sets}}\webright ) \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathord {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\circ \webleft (\mathord {\times }\times \mathord {\times }\webright )\circ \mathbf{\mu }^{\mathsf{Cats}_{\mathsf{2}}}_{4|\mathsf{Sets},\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}\circ \webleft (\webleft (\text{id}_{\mathsf{Sets}}\times \text{id}_{\mathsf{Sets}}\webright )\times \Delta _{\mathsf{Sets}}\webright ) \]

  of Definition 3.3.2.1.1.
• The Left Annihilator. The natural isomorphism
  \[ \zeta ^{\mathsf{Sets}}_{\ell } \colon \mathbb {0}^{\mathsf{Sets}}\circ \mathbf{\mu }^{\mathsf{Cats}_{\mathsf{2}}}_{4|\mathsf{Sets},\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}\circ \webleft (\text{id}_{\mathsf{pt}}\times \mathbf{\epsilon }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}}\webright ) \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\times \circ \webleft (\mathbb {0}^{\mathsf{Sets}}\times \text{id}_{\mathsf{Sets}}\webright ) \]

  of Definition 3.3.3.1.1.

• The Right Annihilator. The natural isomorphism
  \[ \zeta ^{\mathsf{Sets}}_{r} \colon \mathbb {0}^{\mathsf{Sets}}\circ \mathbf{\mu }^{\mathsf{Cats}_{\mathsf{2}}}_{4|\mathsf{Sets},\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}\circ \webleft (\mathbf{\epsilon }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}}\times \text{id}_{\mathsf{pt}}\webright ) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\times \circ \webleft (\text{id}_{\mathsf{Sets}}\times \mathbb {0}^{\mathsf{Sets}}\webright ) \]

  of Definition 3.3.4.1.1.